A beverage is any kind of a drink.
The word "beverage" is often used by hotels, restaurants, and cafes when referring to the different kinds of drinks that are available.
However, you should know that in normal conversation, most Americans just use the word "drink."

Would you like a beverage?

What kind of a beverage would you like to order?

Beverages are listed towards the back of the menu.

There are ice cold beverages in the cooler.

A hot beverage is the thing to have on a cold night.

A cold beverage tastes great on a hot summer day.

The flight attendant pushed a beverage cart through the aisle of the plane and served the passengers.